One has to mention that for the upper range in Fig. REF ,
we used 10k benchmark points (BPs) and considered many theoretical
and experimental constraints such as the vacuum stability, perturbativity,
perturbative unitarity, electroweak precision tests, the di-photon
Higgs decay, the Higgs invisible decay when applicable, the Higgs
total decay width measurement [1]}, [2]},
the implications of the negative searches on neutralinos and charginos
in supersymmetric models on the inert masses, the bounds on DM nucleon
scattering cross section from DD experiments (Xenon 1T [3]});
and the Higgs signal strength at the LHC \(\mu _{{\rm tot}}\ge 0.89\)  [4]}.
For the lower range, we omitted the BPs that are excluded by the negative
searches for a heavy resonance in the channels \(pp\rightarrow S\rightarrow \tau \tau \)  [5]},
\(pp\rightarrow S\rightarrow ZZ\)  [6]} and \(pp\rightarrow S\rightarrow WW\)  [7]};
in addition to the negative searches on the resonant di-Higgs production
\(pp\rightarrow S\rightarrow HH\)  via the different channel \(HH\rightarrow b\bar{b}\tau \tau \)  [8]},
\(HH\rightarrow b\bar{b}b\bar{b}\)  [9]} and \(HH\rightarrow b\bar{b}\gamma \gamma \)  [10]}.
These recent constraints exclude only 5.35% of the BPs used in the
upper panels in Fig. REF . Indeed, there are other relevant
constraints to this model such as neutrinos oscillation data, dark
matter (DM) relic density and and the lepton flavor violating processes.
These constraints are not considered here since we interested on the
parameters and constraints that are relevant to the radiative effects
on the Higgs sector.
<FIGURE>